Evaluate: $\dfrac{3\cos^2 30^\circ - 6\operatorname{cosec}^2 30^\circ}{\tan^2 60^\circ}$
Generated by claude-sonnet-4-6 · 2026-06-15 10:21 · grounding rag
Model Answer
Known values: $\cos 30° = \dfrac{\sqrt{3}}{2}$, $\operatorname{cosec} 30° = 2$, $\tan 60° = \sqrt{3}$
$$= \frac{3\cos^2 30° - 6\operatorname{cosec}^2 30°}{\tan^2 60°}$$
$$= \frac{3\left(\dfrac{\sqrt{3}}{2}\right)^2 - 6(2)^2}{(\sqrt{3})^2}$$
$$= \frac{3 \times \dfrac{3}{4} - 6 \times 4}{3}$$
$$= \frac{\dfrac{9}{4} - 24}{3} = \frac{\dfrac{9 - 96}{4}}{3} = \frac{-87}{4 \times 3} = \frac{-87}{12} = \mathbf{-\dfrac{29}{4}}$$
Source: Chapter 8, Section 8.3 (Trigonometric Ratios of Specific Angles)
---
Explanation
- Substitute standard values directly from Table 8.1: $\cos 30° = \frac{\sqrt{3}}{2}$, $\operatorname{cosec} 30° = 2$, $\tan 60° = \sqrt{3}$.
- Square each value carefully before multiplying by the coefficient.
- Examiners award 1 mark for correct substitution and 1 mark for correct simplification to the final answer $-\dfrac{29}{4}$.
- Avoid arithmetic errors when combining fractions (finding LCM of 4 and 1).